If you look at pages 20 and 21, you will see that at the very top of page 21, it seems to indicate that every sequence that is Cauchy has a limit.

So what is the correct statement? Are all Cauchy sequences convergent? If yes, is the proof in the PDF a good proof? If not, do you have a counter example and why the derivation of the PDF do not work?

I currently suspect that it has something to do with density. Like, we can find Cauchy sequences in that have a limit in .

If you look at pages 20 and 21, you will see that at the very top of page 21, it seems to indicate that every sequence that is Cauchy has a limit.

So what is the correct statement? Are all Cauchy sequences convergent? If yes, is the proof in the PDF a good proof? If not, do you have a counter example and why the derivation of the PDF do not work?

I currently suspect that it has something to do with density. Like, we can find Cauchy sequences in that have a limit in .

Thanks in advance,

This is a property of whatever metric space you're working in. A metric space is called complete if it has the property that every Cauchy sequence converges. For example, and are complete for each . A less trivial example is --the space of all continuous functions equipped with the metric induced by . That said, as you indicated is not complete. There are hunderds of examples I'm sure you could think of (e.g. , , etc.)